The first negative coefficients of symmetric square L-functions

International audienceLet n sym 2 f be the greatest integer such that λ sym 2 f (n) 0 for all n < n sym 2 f and (n, N) = 1, where λ sym 2 f (n) is the nth coefficient of the Dirichlet series representation of the symmetric square L-function L(s, sym 2 f) associated to a primitive form f of level N and of weight k. In this paper we establish the subconvexity bound: n sym 2 f (k 3 N 2) 40/113 where the implied constant is absolute